Q1. A causal discrete-time system is characterized by the difference equation: y[n] = y[n – 2]+ x[n] – x[n – 4]. a. Find the transfer function, H(z), of this system. (Hint: (x* – 1) = (x – 1)(x + 1)(x – j)(x + j)) b. Plot the pole-zero diagram and clearly mention the ROC for H(z). c. Show that the system is unstable by specifying a bounded input, x[n], for which this system results in unbounded output.

Ex. 1472. The plant of a magnetic-gap controller is: Gp(s)=1/((s+f)(s-f)). A controller Gc(s)=K(s+b)(s+c)/s is proposed. Determine the gain K for marginal stability. b=0.50, c=101.00, f=61.30. Determine the gain K for CLGM=10.8 dB. Gp & Gs are in the forward path of unity feedback sys. ans:2

y(n) is output. x(n) is input

3. Consider the system a)  Use state feedback ? = ?? to assign the eigenvalues of ? + ?? at −0.5 ± ?0.5. Plot TT x(t) = [x1(t), x2(t)] for the open- and closed-loop system with x(0) = [−0.6, 0.4] . b)  Design an identity observer with eigenvalues at −? ± ?, where ? > 0. What is the observer gain ? in this case? c)  Use the state estimate ?̂ from (b) in the linear feedback control law ? = ? ?̂, where ? was found in (a). Derive the state-space description of the closed-loop system. If ? = ? ?̂ + ?, what is the transfer function between ? and ?? TT d)  For ?(0) = [−0.6, 0.4] and ?̂(0)= [0, 0] , plot ?(?), ?̂ (?), ?(?), and ?(?) of the closed- loop system obtained in (c) and comment on your results. Use ? = 1, 2, 5, and 10, and comment on the effects on the system response. Remark: This exercise illustrates the deterioration of system response when state observers are used to generate the state estimate that is used in the feedback control law.